<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <a href="csspace3.html#K3" target="_self">Complex_l1_Space</a> ;<br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="csspace3.html#V1" target="_self">CCauchy</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="comseq_1.html#NM1">Complex_Sequence</a> st <br/>( ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Olive">b<sub>2</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <font color="Olive">b<sub>2</sub></font> ) );<br/>
<a NAME="E2:21"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="csspace3.html#E1:21_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  = <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E4:21_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="comseq_1.html#S1">COMSEQ_1:sch 1</a>();<br/></i>
<a NAME="E5:21_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="comseq_2.html#V2">convergent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:21_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>thus </b><a NAME="E2:21_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:21_1_1_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="csspace3.html#E1:21"><i><font color="Green">E16</font></i></a>, <a class="txt" href="csspace3.html#E1:21_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="csspace3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:21_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_1_1_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:21_1_1_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span>
 
<b>by </b><i><a class="ref" href="csspace3.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E3:21_1_1_1_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="csspace3.html#E2:21_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="csspace3.html#E1:21_1_1_1_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E4:21_1_1_1_1_1"/>
 <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="csspace3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:21_1_1_1_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E6:21_1_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:21_1_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<b>hence </b><a NAME="E2:21_1_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:21_1_1_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span>
 
<b>by </b><i><a class="txt" href="csspace3.html#E5:21_1_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rsspace2.html#T4">RSSPACE2:4</a></i>;<br/>
 <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> = 
 <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace3.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T3">CSSPACE:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10">COMSEQ_1:def 10</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace3.html#E4:21_1"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace3.html#E4:21_1"><i><font color="Green">E19</font></i></a></i>
;<br/>
<a NAME="E10:21_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E11:21_1_1_1_1_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="csspace3.html#E3:21_1_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="csspace3.html#E7:21_1_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="comseq_2.html#V2">convergent</a>
 <b>by </b><i><a class="ref" href="comseq_3.html#T46">COMSEQ_3:46</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b><font color="Maroon">c<sub>5</sub></font> =  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>thus </b><a NAME="E6:21_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>] )
 <b>by </b><i><a class="txt" href="csspace3.html#E4:21_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="csspace3.html#E5:21_1"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E4:21"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="csspace3.html#E2:21"><i><font color="Green">E17</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="comseq_1.html#NM1">Complex_Sequence</a> ;<br/>
<div><i><font color="Green">E19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E2:21_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="comseq_1.html#NM1">Complex_Sequence</a> st <br/>( ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Olive">b<sub>2</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="csspace3.html#E4:21"><i><font color="Green">E18</font></i></a></i>;<br/><b>hence </b><a NAME="E3:21_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="comseq_1.html#NM1">Complex_Sequence</a> st <br/>( ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> ) &amp; <font color="Olive">b<sub>1</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E7:21"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:21_3"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E2:21_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="csspace3.html#E1:21_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E3:21_3"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="csspace3.html#E1:21_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:21_3"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="csspace3.html#E1:21"><i><font color="Green">E16</font></i></a>, <a class="txt" href="csspace3.html#E2:21_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="csspace3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E6:21_3"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2 &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_3_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E2:21_3_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="csspace3.html#E5:21_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="csspace3.html#E1:21_3_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E3:21_3_1"/><b>then </b><i><font color="Green">E27</font></i>: 
the <a href="clvect_1.html#U2">norm</a> of <a href="csspace3.html#K3" target="_self">Complex_l1_Space</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>;<br/>
<a NAME="E4:21_3_1"/><i><font color="Green">E28</font></i>: 
 <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="csspace3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:21_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:21_3_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<b>hence </b><a NAME="E2:21_3_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:21_3_1"/>
( <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="csspace3.html#E3:21_3_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="csspace3.html#E4:21_3_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="csspace3.html#D2" target="_self">Def2</a>, <a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:21_3"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st ( for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> ) holds <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:21_3_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds  <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:21_3_2_1"/>
 <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="csspace3.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E2:21_3_2_1"/>
 <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="csspace.html#T3">CSSPACE:3</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> ) holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E2:21_3_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:21_3_2_2_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0
 ;<br/><b>thus </b><a NAME="E2:21_3_2_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2_2_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E2:21_3_2_2_1_1"/><i><font color="Green">E30</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>9</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0 ) &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="csspace3.html#E6:21"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E3:21_3_2_2_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="txt" href="csspace3.html#E1:21_3_2_2_1"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="txt" href="csspace3.html#E1:21_3_2"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p3">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="comseq_1.html#D10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>

;<br/>
<b>hence </b><a NAME="E2:21_3_2_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="txt" href="csspace3.html#E2:21_3_2_2_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b> <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="txt" href="csspace3.html#E2:21_3_2_2_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="csspace3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="txt" href="csspace3.html#E2:21_3_2_2_1_1"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="csspace.html#T3">CSSPACE:3</a></i>
<br/>.= 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E5:21_3_2_2_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0 )
 <b>by </b><i><a class="txt" href="csspace3.html#E2:21_3_2_2_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="csspace3.html#E3:21_3_2_2_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="csspace3.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_3_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:21_3_2"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:21_3_2_3_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> ) holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><b>thus </b><a NAME="E2:21_3_2_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:21_3_2_3_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E3:21_3_2_3_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E4:21_3_2_3_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="csspace3.html#E1:21_3_2_3_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="csspace3.html#E3:21_3_2_3_1_1"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E6:21_3_2_3_1_1"/><i><font color="Green">E34</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>11</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="csspace3.html#E6:21"><i><font color="Green">E19</font></i></a></i>;<br/>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2_3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="csspace3.html#E1:21_3_2_3_1_1"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace3.html#E1:21_3_2"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace3.html#E3:21_3_2_3_1_1"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>

<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace3.html#E6:21_3_2_3_1_1"><i><font color="Green">E34</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>then </b> <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace3.html#E4:21_3_2_3_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="csspace3.html#E6:21_3_2_3_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="csspace3.html#E16"><i><font color="Green">Lemma13</font></i></a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace3.html#E6:21_3_2_3_1_1"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace3.html#E1:21_3_2_3_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="csspace3.html#E3:21_3_2_3_1_1"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T3">CSSPACE:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E9:21_3_2_3_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="csspace3.html#E4:21_3_2_3_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="csspace3.html#E6:21_3_2_3_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="csspace3.html#E7:21_3_2_3_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="csspace3.html#E16"><i><font color="Green">Lemma13</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_3_2_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:21_3_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="csspace3.html#E2:21_3_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="csspace3.html#E3:21_3_2"><i><font color="Green">E29</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:21_3_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) holds <br/>( <font color="Olive">b<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:21_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_3_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:21_3_3"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_3_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E3:21_3_3_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E4:21_3_3_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="csspace3.html#E7:21_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="csspace3.html#E3:21_3_3_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E5:21_3_3_1"/><i><font color="Green">E31</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="csspace3.html#E7:21_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="csspace3.html#E3:21_3_3_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E6:21_3_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_3_3_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:21_3_3_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="csspace3.html#E6:21_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="csspace3.html#E1:21_3_3"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E3:21_3_3_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span>
 
<b>by </b><i><a class="ref" href="rsspace2.html#T4">RSSPACE2:4</a></i>;<br/>
<a NAME="E4:21_3_3_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="csspace3.html#E3:21_3_3_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:21_3_3_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="csspace3.html#E2:21_3_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="csspace3.html#E3:21_3_3_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:21_3_3_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="csspace3.html#E4:21_3_3_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="csspace3.html#E5:21_3_3_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rsspace2.html#T6">RSSPACE2:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:21_3_3"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:21_3_3_2"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E2:21_3_3_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:21_3_3"/><i><font color="Green">E30</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V1">bounded_above</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:21_3_3_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E2:21_3_3_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="csspace3.html#E2:21_3_3"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:21_3_3_3_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="csspace3.html#E3:21_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_3_3_3"/>
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:21_3_3"/><i><font color="Green">E31</font></i>: 
 <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E6:21_3_3"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace3.html#E3:21_3_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="csspace3.html#E4:21_3_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="series_1.html#T20">SERIES_1:20</a></i>;<br/>
<a NAME="E7:21_3_3"/><b>then </b>
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E8:21_3_3"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="csspace3.html#E2:21_3_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rsspace2.html#T6">RSSPACE2:6</a></i>;<br/>
<b>hence </b><a NAME="E9:21_3_3"/>
( <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="csspace3.html#E3:21_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="csspace3.html#E3:21_3_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="csspace3.html#E4:21_3_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="csspace3.html#E5:21_3_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="series_1.html#T20">SERIES_1:20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:21_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:21"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>proof </b></a><div class="add">

<a NAME="E1:21_4"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:21_4_1"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E2:21_4_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:21_4"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="csspace3.html#E7:21"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E4:21_4"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace3.html#E3:21_4"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E5:21_4"/>
 <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="csspace3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:21_4"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span>;<br/>
<a NAME="E7:21_4"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K9">+</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace3.html#E4:21_4"><i><font color="Green">E23</font></i></a>, <a class="txt" href="csspace3.html#E6:21_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="series_1.html#T10">SERIES_1:10</a></i>;<br/>
<a NAME="E8:21_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K9">+</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<i><font color="Green">E26</font></i>: <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p4">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>

;<br/>
<a NAME="E2:21_4_2"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>
<a NAME="E3:21_4_2"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>
<a NAME="E4:21_4_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="csspace3.html#E1:21_4_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="csspace3.html#E2:21_4_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="complex1.html#T145">COMPLEX1:145</a></i>;<br/>
<a NAME="E5:21_4_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E6:21_4_2"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K9">+</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:21_4"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="txt" href="csspace3.html#E1:21_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="csspace3.html#E7:21_4"><i><font color="Green">E25</font></i></a>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:21"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font> is <a href="comseq_3.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E10:21"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K1">the_set_of_ComplexSequences</a> 
 
<b>by </b><i><a class="ref" href="csspace.html#D1">CSSPACE:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="csspace3.html#K3" target="_self">Complex_l1_Space</a>  <b>by </b><i><a class="txt" href="csspace3.html#E9:21"><i><font color="Green">E21</font></i></a>, <a class="ref" href="csspace3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:21"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:21_5"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:21_5"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>( <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="csspace3.html#E7:21"><i><font color="Green">E20</font></i></a>, <a class="txt" href="csspace3.html#E1:21_5"><i><font color="Green">E23</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:21_5_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace3.html#K3" target="_self">Complex_l1_Space</a>  <b>by </b><i><a class="txt" href="csspace3.html#E9:21"><i><font color="Green">E21</font></i></a>, <a class="txt" href="csspace3.html#E10:21"><i><font color="Green">E22</font></i></a>, <a class="ref" href="csspace3.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:21_5_1"/><i><font color="Green">E26</font></i>: 
 <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="csspace3.html#E3:21_5"><i><font color="Green">E24</font></i></a>, <a class="txt" href="csspace3.html#E1:21_5_1"><i><font color="Green">E25</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace3.html#K3" target="_self">Complex_l1_Space</a>  ;<br/><a NAME="E5:21_5_1"/>
 <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="csspace3.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E6:21_5_1"/><b>then </b>
 <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="comseq_1.html#K9">.|</a></span>
 <b>by </b><i><a class="ref" href="csspace3.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E7:21_5_1"/><b>then </b><i><font color="Green">E27</font></i>: 
the <a href="clvect_1.html#U2">norm</a> of <a href="csspace3.html#K3" target="_self">Complex_l1_Space</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="csspace3.html#E3:21_5_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="csspace3.html#D2" target="_self">Def2</a></i>;<br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47">RLVECT_1:47</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#T104">CLVECT_1:104</a></i>
;<br/><b>hence </b><a NAME="E9:21_5_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="csspace3.html#E7:21_5_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:21_5"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:21"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="clvect_1.html#V6">convergent</a>
 <b>by </b><i><a class="ref" href="clvect_1.html#D15">CLVECT_1:def 15</a></i>;<br/>


</div>
